
If \(d_1=d_2\)  and \(\Psi (y)=\frac{1}{2} |y|^2\) , i.e. the measure \(\mu _2\)  is the standard Gaussian measure on \(\mathbb {R}^{d_2}\) , then thm:ess-m-diss is a slight improvement of [1]} due to a better trade-off between \(\Sigma \)  and \(\Phi \) . In that case, we get \(\alpha _2\equiv c_\Sigma \)  in (REF ). In particular, the convergence is exponential if there is a (strong) Poincaré inequality for \(\Phi \) .

If \(\Sigma =I\) , then thm:main-result yields the same convergence result as [2]}.
However, we show that \(\mu _2(|\nabla _y\Psi |^4)<\infty \)  need not be assumed and give an alternative condition for \(\Psi \)  by replacing
\(\sup _{r\ge 0} \left| \psi ^{\prime }(r)+2r\psi ^{\prime \prime }(r)-\frac{2r\psi ^{\prime \prime \prime }(r)+(d_2+2)\psi ^{\prime \prime }(r)}{\psi ^{\prime }(r)} \right| < \infty \) 
with \(L^2(\mu _2)\) -integrability of the third-order derivatives of \(\Psi \) .

Since the convergence rate \(\xi \)  depends from the choice of \(\Sigma \)  only via the ellipticity constant \(c_\Sigma \)  as a factor in front of \(\alpha _2\) , we refer to [2]} for a detailed list of explicit rates depending on the chosen potentials. We list two examples for illustration purposes:

Let \(\Phi (x)=k(1+|x|^2)^{\delta /2}+g_1(x)\)  and \(\Psi (y)=\kappa (1+|y|^2)^{\varepsilon /2}+g_2(y)\) ,
where \(k,\kappa ,\delta ,\varepsilon >0\)  are constant and \(g_1\in C_b^2(\mathbb {R}^{d_1})\) , \(g_2 \in C_b^3(\mathbb {R}^{d_2})\)  with \(g_2(y)=\eta (|y|^2)\)  for some \(\eta \in C^3([0,\infty ))\) .
Then, (REF ) holds with
\(\xi (t)=\exp (-c_2 t^{\omega (\delta ,\varepsilon )}),\text{ where }\omega (\delta ,\varepsilon )=\frac{\delta \varepsilon }{\delta \varepsilon +8\varepsilon (1-\delta )^{+}+4\delta (1-\varepsilon )^{+}},\) 
for some constants \(0<c_1,c_2<\infty \) . In particular, if \(\delta ,\varepsilon \ge 1\) , then the decay rate is exponential.

Let \(d\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =d1=d2\(, \) (x)=q+d2 (1+|x|2)+g1(x)\( for some \) q>0\( and\) (y)=p+d2(1+|y|2)+g2(y)\( for some \) p>0\(,where \) g1Cb2(Rd)\( and \) g2 Cb3(Rd)\(with \) g2(y)=(|y|2)\( for some \) C3([0,))\(.Then, (\ref {eq:conv-estimate}) holds with\)\(\begin{aligned}\xi (t) &= c_2(1+t)^{-\omega (p,q)}(\log (\mathrm {e}+t))^{\omega (p,q)},&& \text{ where }\\\omega (p,q) &= \frac{1}{2\theta (q)+\theta (p)+2\theta (q)\theta (p)}&& \text{ and }\\\theta (r) &= \frac{d+r+2}{r} \wedge \frac{4r+4+2d}{(r^2-4-2d-2r)^{+}},\end{aligned}\)\(for some constants \) 0<c1,c2<\(.\)

Essential m-dissipativity of the operator \(L\)
Preliminaries
Throughout this section, we assume ass:ellipticity-ass:coeff-growth, ass:y-potential-1-ass:y-potential-3 and ass:x-potential-1-ass:x-potential-2 to be satisfied.
Definition 2.1 
Depending on the choice of \(\Sigma \) , we introduce the following constants:

M={ aij: 1i,j d2 },
B={ |j aij(y)|: yB1(0), 1i,j d2 }        and 
N=M2+B2+d2M2.

We introduce the following useful decomposition of \(L\) , which we require later to apply the weak hypocoercivity method:
Definition 2.2 
Let \(\mathcal {C}\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =Cc(Rd1)Cc(Rd2)\( be the tensor product space of smooth compactly supported functions on \) Rd1\( and \) Rd2\(.We define the linear operators \) S\( and \) A\( for \) fC\( by\begin{align*}Sf &\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\end{align*}=\sum _{i,j=1}^{d_2} a_{ij}\partial _{y_j}\partial _{y_i}f + \sum _{i=1}^{d_2} b_i\partial _{y_i}f, \\&\quad \text{ where } b_i(y)= \sum _{j=1}^{d_2}(\partial _j a_{ij}(y)-a_{ij}(y)\partial _j\Psi (y)),\\Af &\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =Q*y f - Qx f.
Integration by parts shows that \((S,\mathcal {C})\)  is symmetric and non-positive definite on \(X\) , and \((A,\mathcal {C})\)  is antisymmetric on \(X\) , which implies both are dissipative and therefore closable.
We denote their closure respectively by \((S,D(S))\)  and \((A,D(A))\) 
Since \((L,\mathcal {C})=(S-A,\mathcal {C})\) , it is also dissipative and closable, and we denote its closure by \((L,D(L))\) .

We observe that for \(f\in \mathcal {C}\)  and \(g\in H^{1,2}(E,\mu )\) , integration by parts yields
\((Lf,g)_X = -\int _E \left\langle \nabla f, \begin{pmatrix}0 & -Q\\Q^* & \Sigma \end{pmatrix}\nabla g \right\rangle \, \mathrm {d}\mu .\) 
We also provide the following \(L^p\) -integrability result, which is based on and vastly generalizes a Lemma from Villani, see [4]}
Theorem 2.3 
Let \(V\in C^1(\mathbb {R}^d)\cap H_{\mathrm {loc}}^{2,\infty }(\mathbb {R}^d)\)  satisfy ass:y-potential-3 with \(\Psi =V\)  and let \(k\in \mathbb {N}\) .
Then there is a constant \(C_k<\infty \)  such that for all \(g\in H^{1,2k}(\mathrm {e}^{-V}\mathrm {d}x)\) ,
the following holds:
\(\int _{\mathbb {R}^d}|\nabla V|^{2k}g^{2k}\mathrm {e}^{-V}\,\mathrm {d}x\le C_k\left( \int _{\mathbb {R}^d}g^{2k}\mathrm {e}^{-V}\,\mathrm {d}x+ \int _{\mathbb {R}^d}|\nabla g|^{2k}\mathrm {e}^{-V}\,\mathrm {d}x \right).\) 


Proof 1 The proof is derived from the proof of [4]}, which shows the statement for \(\alpha =1\)  and \(k=1\) . We therefore assume \(\alpha >1\) , but also only show the proof for \(k=1\) , as the other cases follow analogously after adjusting the constant.
Without loss of generality, we assume that \(g\in C_c^\infty (\mathbb {R}^{d})\) , since that space is dense in \(H^{1,2k}(\mathrm {e}^{-V(x)}\,\mathrm {d}x)\)  (which can be proven analogously to the well-known denseness result in \(H^{m,p}(\Omega )\)  for \(\Omega \)  satisfying the segment property, since \(\mathrm {e}^{-V(x)}\,\mathrm {d}x\)  is locally equivalent to the Lebesgue measure).
Let the measure \(\mathrm {e}^{-V}\mathrm {d}x\)  be denoted by \(\mu _V\) . Then, integration by parts yields
\(\int _{\mathbb {R}^d}|\nabla V|^2g^2\,\mathrm {d}\mu _V= -\int _{\mathbb {R}^d}g^2\langle \nabla V, \nabla (\mathrm {e}^{-V})\rangle \,\mathrm {d}x= \int _{\mathbb {R}^d}g^2\Delta V \,\mathrm {d}\mu _V+ \int _{\mathbb {R}^d}2g\langle \nabla V, \nabla g\rangle \,\mathrm {d}\mu _V.\) 
Due to ass:y-potential-3, it holds that
\(|\Delta V|=\sqrt{(\Delta V)^2}\le \sqrt{d}\,|\nabla ^2 V|\le \sqrt{d}K(1+|\nabla V|^\alpha ).\) 
This with Cauchy–Bunyakovsky–Schwarz implies
\(\int _{\mathbb {R}^d}g^2\Delta V \,\mathrm {d}\mu _V\le \sqrt{d}K\left(\int _{\mathbb {R}^d}g^2\,\mathrm {d}\mu _V+\sqrt{\int _{\mathbb {R}^d}g^2|\nabla V|^2 \,\mathrm {d}\mu _V}\sqrt{\int _{\mathbb {R}^d}g^2|\nabla V|^{2(\alpha -1)} \,\mathrm {d}\mu _V} \right),\) 
and Young's inequality further gives
\(\int _{\mathbb {R}^d}g^2\Delta V \,\mathrm {d}\mu _V\le \sqrt{d}K\int _{\mathbb {R}^d}g^2\,\mathrm {d}\mu _V+\frac{1}{4} \int _{\mathbb {R}^d}|\nabla V|^2g^2\,\mathrm {d}\mu _V+dK^2 \int _{\mathbb {R}^d}|\nabla V|^{2(\alpha -1)}g^2\,\mathrm {d}\mu _V.\) 
Similarly, it holds that
\(\begin{aligned}2\int _{\mathbb {R}^d}g\langle \nabla V, \nabla g\rangle \,\mathrm {d}\mu _V&\le 2\sqrt{\int _{\mathbb {R}^d}g^2|\nabla V|^2 \,\mathrm {d}\mu _V}\sqrt{\int _{\mathbb {R}^d}|\nabla g|^2 \,\mathrm {d}\mu _V}\\&\le \frac{1}{4} \int _{\mathbb {R}^d}|\nabla V|^2g^2\,\mathrm {d}\mu _V+4\int _{\mathbb {R}^d}|\nabla g|^2 \,\mathrm {d}\mu _V.\end{aligned}\) 
Adding these estimates together and subtracting \(\frac{1}{2}\int _{\mathbb {R}^{d}} |\nabla V|^2g^2\,\mathrm {d}\mu _V\)  from both sides, we obtain
\(\int _{\mathbb {R}^d}|\nabla V|^2g^2\,\mathrm {d}\mu _V\le 2\sqrt{d}K\int _{\mathbb {R}^d}g^2\,\mathrm {d}\mu _V+8\int _{\mathbb {R}^d}|\nabla g|^2 \,\mathrm {d}\mu _V+2dK^2\int _{\mathbb {R}^d}|\nabla V|^{2(\alpha -1)}g^2\,\mathrm {d}\mu _V.\) 
If \(2(\alpha -1)\le 1\) , then \(|\nabla V|^{2(\alpha -1)}\le 1+|\nabla V|\) , and therefore
\(2dK^2\int _{\mathbb {R}^d}|\nabla V|^{2(\alpha -1)}g^2\,\mathrm {d}\mu _V\le 2dK^2\int _{\mathbb {R}^d}g^2\,\mathrm {d}\mu _V+2d^2K^4 \int _{\mathbb {R}^d}g^2\,\mathrm {d}\mu _V+\frac{1}{2} \int _{\mathbb {R}^d}|\nabla V|^2g^2\,\mathrm {d}\mu _V,\) 
hence
\(\int _{\mathbb {R}^d}|\nabla V|^2g^2\,\mathrm {d}\mu _V\le 4(\sqrt{d}K+dK^2+d^2K^4)\int _{\mathbb {R}^d}g^2\,\mathrm {d}\mu _V+16\int _{\mathbb {R}^d}|\nabla g|^2 \,\mathrm {d}\mu _V,\) 
which satisfies the claimed inequality.
In the remaining case that \(2(\alpha -1)\in (1,2)\) , we obtain instead
\(2dK^2\int _{\mathbb {R}^d}|\nabla V|^{2(\alpha -1)}g^2\,\mathrm {d}\mu _V\le 2d^2K^4\int _{\mathbb {R}^d}|\nabla V|^{2(2(\alpha -1)-1)}g^2\,\mathrm {d}\mu _V+\frac{1}{2} \int _{\mathbb {R}^d}|\nabla V|^2g^2\,\mathrm {d}\mu _V.\) 
Repeating this procedure iteratively provides us with estimates of the form
\(\begin{aligned}\int _{\mathbb {R}^d}|\nabla V|^2g^2\,\mathrm {d}\mu _V&\le 2^{m+2}\int _{\mathbb {R}^d}|\nabla g|^2 \,\mathrm {d}\mu _V+2^{m}\sqrt{d}K\int _{\mathbb {R}^d}g^2\,\mathrm {d}\mu _V\\&\qquad +(2dK^2)^{2^{(m-1)}}\int _{\mathbb {R}^d}|\nabla V|^{\gamma _m(\alpha )}g^2\,\mathrm {d}\mu _V,\end{aligned}\) 
where \(\gamma _m(\alpha )=2^m(\alpha -2)+2\) .
Since there is some \(\varepsilon >0\)  such that \(\alpha =2-\varepsilon \) ,
we can write \(\gamma _m(\alpha )=2-2^m\varepsilon \) , so there is some \(m_0\in \mathbb {N}\)  such that \(0<\gamma _{m_0}(\alpha )\le 1\) . Then the claim follows as above for the case \(2(\alpha -1)\le 1\) .

Remark 2.4 
The conditions ass:y-potential-1–ass:y-potential-3 together imply that \(\Psi \in C^1(\mathbb {R}^{d_2})\)  and that \(\nabla \Psi \)  is locally Lipschitz-continuous. This can be proven using Sobolev embedding and interpolation theory, see [6]} or [7]}. Therefore, we can apply thm:arbitrary-integrability with \(g\equiv 1\)  to obtain \(|\nabla \Psi |\in L^p(\mu _2)\)  for all \(1\le p<\infty \) . Due to ass:y-potential-3, this also implies \(|\nabla ^2\Psi |\in L^p(\mu _2)\)  for all \(1\le p<\infty \) , where \(\nabla ^2\Psi \)  denotes the Hessian matrix.


Essential m-dissipativity for Lipschitz coefficients and potential
We begin with an essential self-adjointness result for the symmetric part \((S,\mathcal {C})\) .
Since none of the coefficients depend on the coordinate \(x\) , we first reduce it to an operator on \(L^2(\mathbb {R}^{d_2};\mu _2)\) .
Lemma 2.5 
The differential operator \((\tilde{S},C_c^\infty (\mathbb {R}^{d_2}))\) , where
\(\tilde{S}f=\sum _{i,j=1}^d a_{ij}\partial _j\partial _if + \sum _{i=1}^d \sum _{j=1}^d(\partial _j a_{ij}-a_{ij}\partial _j\Psi )\partial _if \quad \text{ for }f\in C_c^\infty (\mathbb {R}^{d_2}),\) 
is essentially self-adjoint on \(L^2(\mathbb {R}^{d_2},\mu _2)\) .

Proof 2 We apply [1]} with \(A=\Sigma \)  and \(\rho =\mathrm {e}^{-\Psi }\)  to show essential self-adjointness. For this we need \(\mathrm {e}^{\Psi }\)  to be locally bounded, which is satisfied due to ass:y-potential-1. The conditions on \(A\)  are satisfied due to ass:ellipticity–ass:coeff-derivatives, and sufficient regularity of \(\rho \)  follows from rem:psi-lp.

Corollary 2.6 The symmetric operator \((S,\mathcal {C})\)  as defined in def:operators is essentially self-adjoint on \(X=L^2(E;\mu )\) .

Proof 3 let \(g=g_1\otimes g_2\in \mathcal {C}\)  be a pure tensor.
Then, due to lem:ess-self-adjoint-y, there is a sequence \((\tilde{f}_n)_{n\in \mathbb {N}}\)  in \(C_c^\infty (\mathbb {R}^{d_2})\)  such that \((I-\tilde{S})\tilde{f}_n\rightarrow g_2\)  in \(L^2(\mathbb {R}^{d_2},\mu _2)\)  as \(n\rightarrow \infty \) . Define \(f_n \in \mathcal {C}\)  for each \(n\in \mathbb {N}\)  as \(f_n\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =g1fn\(.Then\)\(\Vert (I-S)f_n-g\Vert _X= \Vert  g_1\otimes ((I-\tilde{S})\tilde{f}_n - g_2)\Vert _X= \Vert g_1\Vert _{L^2(\mu _1)}\cdot \Vert (I-\tilde{S})\tilde{f}_n-g_2 \Vert _{L^2(\mu _2)},\)\(which converges to zero as \) n\(.By taking linear combinations, this shows that \) (I-S)(C)\( is dense in \) X\(, since \) C\( is dense in \) X\(. Therefore, the dissipative operator \) (S,C)\( is essentially self-adjoint.\)
Since \(S\)  is dissipative on \(D_0\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =Lc2(1)Cc(Rd2)C\(,the operator \) (S,D0)\( is essentially m-dissipative as well. We introduce the unitary transformations\begin{align}U:H &\rightarrow L^2(E,\mathrm {d}(x,y)), && f\mapsto \sqrt{Z(\Psi )^{-1}}\,\mathrm {e}^{-\frac{1}{2}(\Phi +\Psi )} \qquad \text{ and }\\U_\Psi : L^2(\mu _2) &\rightarrow L^2(\mathbb {R}^{d_2},\mathrm {d}y), && f\mapsto \sqrt{Z(\Psi )^{-1}}\,\mathrm {e}^{-\frac{1}{2}\Psi }\end{align}as well as the subspace \) D1=Lc2(Rd1,dx)UCc(Rd2)\( of \) L2(E,d(x,y))\(.Note that due to {ass:x-potential-1}, \) e-\( is strictly positive and locally bounded, which implies that \) Lc2(Rd1,dx)\( and \) Lc2(1)\( coincide. Hence we obtain \) D1=UD0\( and essential m-dissipativity of \) (L0,D1)\(, where \) L0=USU-1\(. For \) fD1\(, we obtain the representation\)\(L_0f = \sum _{i,j=1}^d a_{ij}\partial _{y_j}\partial _{y_i}f- \frac{1}{4}(\nabla \Psi ,\Sigma \nabla \Psi )f+ \frac{1}{2} \sum _{i,j=1}^{d_2} a_{ij}\partial _j\partial _i\Psi f+ \sum _{i,j=1}^d \partial _j a_{ij} (\tfrac{1}{2}\partial _i\Psi f+\partial _{y_i}f),\)\(where the differential operators \)\( and \) i\( are understood in the distributional sense.\) Next we perturb \((L_0,D_1)\)  by the multiplication operator \((A_1,D_1)\)  given by the measurable function \(\mathrm {i}Q\nabla \Psi \cdot x:E\rightarrow .Clearly \) (A1,D1)\( is well-defined in \) L2(E,d(x,y))\( and antisymmetric, hence dissipative.\begin{prop}The operator (L_1,D_1) defined by L_1\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\end{prop}=L_0+A_1\)  is essentially m-dissipative on \( L^2(E,\mathrm {d}(x,y))\) .
Proof 4 We introduce the complete orthogonal family of projections \(P_n\)  defined via \(P_nf\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =nf\(,where \) n\( is given by \) n(x,y) = 1[n-1,n)(|x|)\(.Each \) Pn\( leaves \) D1\( invariant and commutes with both \) L0\( and \) A1\(.We have to show that each \) PnA1\( is \) PnL0\(-bounded with Kato bound zero. Let \) fPnD1\(. Then it holds that\)\(\Vert \mathrm {i}Q\nabla \Psi \cdot xf\Vert _{L^2}^2\le n^2|Q|_2^2\int _E |\nabla \Psi |^2f^2\,\mathrm {d}(x,y).\)\(Hence, it is enough to show that there are finite constants \) a,b\( such that\begin{equation}\int _E |\nabla \Psi |^2f^2\,\mathrm {d}(x,y)\le a(L_0f,f)+b\Vert f\Vert _{L^2}\quad \text{ for all }f\in D_1.\end{equation}We get\)\(\begin{aligned}\int _E |\nabla \Psi |^2f^2\,\mathrm {d}(x,y) &\le 4c_\Sigma \int _E \frac{1}{4}\langle \nabla \Psi ,\Sigma \nabla \Psi \rangle f^2\,\mathrm {d}(x,y) \\&\le 4c_\Sigma \left( \int _E \frac{1}{4}\langle \nabla \Psi ,\Sigma \nabla \Psi \rangle f^2\,\mathrm {d}(x,y)+ \int _E \langle \nabla _y f,\Sigma \nabla _yf\rangle \,\mathrm {d}(x,y) \right) \\&= 4c_\Sigma \left( (-L_0f,f)_{L^2}+\int _E\frac{1}{2} \sum _{i,j=1}^{d_2} (a_{ij}\partial _j\partial _i\Psi + \partial _j a_{ij} \partial _i\Psi ) f^2\,\mathrm {d}(x,y) \right)\end{aligned}\)\(Let \) R1=4c\( and recall that due to {ass:coeff-growth} with \) =0\(, it holds that \) |j aij|2M\(. Using the Hölder and Young inequalities for \) p=q=2\(, it follows that\)\(\begin{aligned}\frac{R_1}{2}\left|\int _E \sum _{i,j=1}^{d_2} \partial _j a_{ij}\partial _i\Psi f^2\,\mathrm {d}(x,y)\right|&\le \frac{R_1}{2}\sum _{i=1}^{d_2}\left\Vert  \left(\sum _{j=1}^{d_2}\partial _j a_{ij} \right)fR_1^{1/2} \right\Vert _{L^2} \cdot \Vert R_1^{-1/2}\partial _i\Psi f\Vert _{L^2} \\&\le \frac{1}{4}\sum _{i=1}^{d_2}\int _E \left(R_1^2\Biggl (\sum _{j=1}^{d_2}\partial _j a_{ij}\Biggr )^2+(\partial _i\Psi )^2 \right) f^2\,\mathrm {d}(x,y) \\&\le 16c_\Sigma ^2M^2d_2^3\Vert f\Vert _{L^2}^2+\frac{1}{4} \int _E |\nabla \Psi |^2f^2\,\mathrm {d}(x,y).\end{aligned}\)\(\) Now recall ass:y-potential-3 and set \(R_2\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =8cMK\(; then again with Hölder and Young, but for \) p=2\(, \) q=22-\(, we get\)\(\begin{aligned}\frac{R_1}{2}\left|\int _E \sum _{i,j=1}^{d_2} a_{ij}\partial _j\partial _i\Psi f^2\,\mathrm {d}(x,y)\right|&\le \frac{R_1}{2}\int _E |\Sigma |_2 \cdot |\nabla ^2\Psi |_2f^2\,\mathrm {d}(x,y) \\&\le 2c_\Sigma M_\Sigma K \int _E (1+|\nabla \Psi |^\alpha ) R_2^{-\frac{\alpha }{2}} f^\alpha R_2^{\frac{\alpha }{2}}f^{2-\alpha }\,\mathrm {d}(x,y) \\&\le \frac{R_2}{4}\left( \Vert f\Vert _{L^2}^2+ \Vert  |\nabla \Psi |^\alpha R_2^{-\frac{\alpha }{2}} f^\alpha \Vert _{L^{\frac{2}{\alpha }}} \cdot \Vert  R_2^{\frac{\alpha }{2}}f^{2-\alpha } \Vert _{L^{\frac{2}{2-\alpha }}} \right) \\&\le \frac{R_2}{4} \Vert f\Vert _{L^2}^2+ \frac{\alpha R_2}{8 R_2}\Vert  |\nabla \Psi |f\Vert _{L^2}^2+\frac{(2-\alpha )R_2^{\frac{2}{2-\alpha }}}{8}\Vert f\Vert _{L^2}^2 \\&\le \left(\frac{R_2}{4}+\frac{R_2^{\frac{2}{2-\alpha }}}{8} \right)\Vert f\Vert _{L^2}^2 +\frac{1}{4} \int _E |\nabla \Psi |^2f^2\,\mathrm {d}(x,y).\end{aligned}\)\(Combining these three inequalities yields (\ref {eq:Kato-bound}) with\)\(a=-8c_\Sigma \quad \text{ and }\quad b=2R_1^2M^2d_2^3+\frac{R_2}{2}+\frac{R_2^{\frac{2}{2-\alpha }}}{4}.\)\(\)
Since \(C_c^\infty (\mathbb {R}^{d_1})\otimes U_\Psi C_c^\infty (\mathbb {R}^{d_2})\)  is dense in \(D_1\)  wrt. the graph norm of \(L_1\) ,
we obtain essential m-dissipativity of \((L_1,C_c^\infty (\mathbb {R}^{d_1})\otimes U_\Psi C_c^\infty (\mathbb {R}^{d_2}))\)  and therefore also of its dissipative extension \((L_1,D_2)\)  with \(D_2\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =S(Rd1)UCc(Rd2))\(,where \) S(Rd1)\( denotes the set of smooth functions of rapid decrease on \) Rd1\(.Applying Fourier transform in the \) x\(-component leaves \) D2\( invariant and shows that\) (L2,D2)\( is essentially m-dissipative, where \) L2= L0+Qx\(.Now we add the part depending on the potential \)\(.\)
Proposition 2.7 Let \(\Sigma \)  satisfy ass:coeff-growth with \(\beta =0\)  and let \(\Phi \)  be Lipschitz-continuous.
Then the operator \((L^{\prime },D_2)\)  with \(L^{\prime }=L_2-Q^*\nabla \Phi \nabla _y\)  is essentially m-dissipative on \(L^2(E,\mathrm {d}(x,y))\) .

Proof 5 It holds due to antisymmetry of \(Q\nabla \Psi \nabla _x\)  that
\(\Vert Q^*\nabla \Phi \nabla _yf\Vert _{L^2}^2 \le \Vert |Q^*\nabla \Phi |\Vert _\infty ^2c_\Sigma \biggl ((\nabla _yf,\Sigma \nabla _y f)_{L^2}+ \Bigl (\frac{\langle \nabla \Psi ,\Sigma \nabla \Psi \rangle }{4} f -Q\nabla \Psi \nabla _x f,f\Bigr )_{L^2} \biggr ),\) 
which analogously to the proof of prop:no-potential again implies
that the antisymmetric, hence dissipative operator \((\nabla \Phi \nabla _y,D_2)\)  is \(L_2\) -bounded with bound zero.
This shows the claim.

Denote by \(H_c^{1,\infty }(\mathbb {R}^{d_1})\)  the space of functions in \(H^{1,\infty }(\mathbb {R}^{d_1},\mathrm {d}x)\)  with compact support
and set \(D^{\prime }\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =Hc1,(Rd1)UCc(Rd2)\(. As \) (L',D')\( is dissipative and its closure extends \) (L',D2)\(, it is itself essentially m-dissipative. The unitary transformation \) U-1\( from the beginning of this section transforms \) D'\( into \) D3=Hc1,(Rd1)Cc(Rd2)\(, and it holds that \) U-1L'U = L\( on \) D3\(.Since \) ||Lloc2(Rd2,2)\(, by the same argument as in \cite [Theorem 4.1]{BG21} we get the following result:\)
Theorem 2.8 
Let \(\Sigma \)  satisfy ass:coeff-growth with \(\beta =0\)  and \(\Phi \)  be Lipschitz-continuous.
Then \((L,\mathcal {C})\)  is essentially m-dissipative on \(H\) .

Essential m-dissipativity for locally Lipschitz coefficients and potential
Now we extend the above result to the general statement given in thm:ess-m-diss.
The proof is an adaptation of the procedure employed for [1]}.
Without loss of generality assume \(\Phi \ge 0\) .
For \(n\in \mathbb {N}\)  we define \(\Sigma _n\)  via
\(\Sigma _n=(a_{ij,n})_{1\le i,j\le d}, \quad a_{ij,n}(y)\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) 
=aij((n|y|1)y).
\(Then each \) 
n\( coincides with \)\( on \) Bn(0)\( and satisfies {ass:ellipticity} with \) cn=c\(.For \) yBn(0)\(, {ass:coeff-growth} implies \) |k aij,n|2Mn\( for all \) 1kd2\(.For \) yRd2Bn(0)\(, the chain rule suggests\)\(|\partial _ka_{ij,n}|=\left| \partial _ka_{ij}\left(\frac{yn}{|y|}\right)-\sum _{\ell =1}^{d_2}\partial _\ell a_{ij}\left(\frac{yn}{|y|}\right)\frac{ny_k y_\ell }{|y|^3} \right|\le (\sqrt{d_2}+1)Mn^\beta .\)\(Hence, \) n\( satisfies {ass:coeff-growth} with \) n= 0\( and \) Mn=(d2+1)Mn\(.\) Let further \(\eta _m\in C_c^\infty (\mathbb {R}^{d_1})\)  for each \(m\in \mathbb {N}\)  with \(\eta =1\)  on \(B_m(0)\)  and set \(\Phi _m=\eta _m \Phi \) , which is Lipschitz-continuous. Then set \(\mu _{1,m}\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =e-mdx\(, \) Xm =L2(E,1,m2)\( and define \) (Ln,m,C)\( via\)\(L_{n,m}f = \sum _{i,j=1}^d a_{ij,n}\partial _{y_j}\partial _{y_i}f + \sum _{i=1}^d \sum _{j=1}^d(\partial _j a_{ij,n}-a_{ij,n}\partial _j\Psi )\partial _{y_i}f + Q\nabla \Psi \cdot \nabla _x f - Q^*\nabla \Phi _m\cdot \nabla _y f.\)\(Then by {thm:generator-nice-coeff}, for each \) n,mN\(, \) (Ln,m,C)\( is essentially m-dissipative on \) Xm\(,and it holds that \) Ln,mf = Lf\( for all \) fC\( on \) Bm(0)Bn(0)\(.Note further that \) XXm\(.\) Now we prove the analogue to [1]}:
Lemma 2.9 
Let \(n,m\in \mathbb {N}\)  and \(\Sigma _n\) , \(\Phi _m\)  as defined above. Then there is a constant \(D_1<\infty \) 
independent of \(n,m\)  such that for each \(1\le j\le d_2\) , the following hold for all \(f\in \mathcal {C}\) :

jfXm D1 n2 (I-Ln,m)f Xm,
yjfXm D1 n2 (I-Ln,m)f Xm.

Proof 6 Define the unitary transformations \(U_m:X_m\rightarrow L^2(E,\mathrm {d}(x,v))\)  analogously to (),
as well as the operator \(L_{n,m}^{\prime }=U_mL_{n,m}U_m^{-1}\) ,
and let \(f\in U_m \mathcal {C}=C_c^\infty (\mathbb {R}^{d_1})\otimes U_\Psi C_c^\infty (\mathbb {R}^{d_2})\) . Then
\(\begin{aligned}L_{n,m}^{\prime }f= \sum _{i,j=1}^d a_{ij,n}\partial _{y_j}\partial _{y_i}f&- \frac{1}{4}(\nabla \Psi ,\Sigma _n \nabla \Psi )f+ \frac{1}{2} \sum _{i,j=1}^{d_2} a_{ij,n}\partial _j\partial _i\Psi f\\&+ \sum _{i,j=1}^d \partial _j a_{ij,n} (\tfrac{1}{2}\partial _i\Psi f+\partial _{y_i}f)-Q\nabla \Psi \cdot \nabla _xf+Q^*\nabla \Phi _m\cdot \nabla _yf.\end{aligned}\) 
Analogously to the proof of prop:no-potential
and due to antisymmetry of \(Q\nabla \Psi \nabla _x\)  and \(Q^*\nabla \Phi _m\nabla _v\)  on \(L^2(\mathrm {d}(x,y))\) , it holds that
\(\begin{aligned}\Vert  \partial _j\Psi U_m^{-1}f\Vert _{X_m}^2 &= \Vert \partial _j\Psi f\Vert _{L^2(\mathrm {d}(x,v))}^2 \le 4 c_{\Sigma }\int _E\frac{1}{4} \langle y,\Sigma _n y\rangle f^2 + \langle \nabla _y f,\Sigma _n\nabla _y f\rangle \,\mathrm {d}(x,v)\\&\le a(-L_{n,m}^{\prime }f,f)_{L^2}+b_n\Vert f\Vert _{L^2}^2,\end{aligned}\) 
where \(a=-8c_\Sigma \)  and
\(b_n= 2R_1^2(M_n)^2d_2^3+\frac{R_2}{2}+\frac{R_2^{\frac{2}{2-\alpha }}}{4}\le 2R_1^24M^2n^{2\beta }d_2^4+\frac{R_2}{2}+\frac{R_2^{\frac{2}{2-\alpha }}}{4}.\) 
Since by the Hölder and Young inequalities, combined with dissipativity of \((L_{n,m}^{\prime },U_m\mathcal {C})\) , it holds that
\(\begin{aligned}(-L_{n,m}^{\prime }f,f)_{L^2}+\Vert f\Vert _{L^2}^2&= ((I-L_{n,m}^{\prime })f,f)_{L^2} \le \frac{1}{4} \left(\Vert (I-L_{n,m}^{\prime })f \Vert _{L^2}+\Vert f\Vert _{L^2}\right)^2 \\&\le \frac{1}{4} \left( 2\Vert (I-L_{n,m}^{\prime })f \Vert _{L^2}\right)^2 = \Vert (I-L_{n,m}^{\prime })f \Vert _{L^2}^2,\end{aligned}\) 
the estimate (REF ) implies the existence of some \(D_1<\infty \)  such that
\(\Vert  \partial _j\Psi U_m^{-1}f\Vert _{X_m}\le D_1n^{\beta }\Vert (I-L_{n,m}^{\prime })f \Vert _{L^2}=D_1n^{\beta }\Vert (I-L_{n,m})U_m^{-1}f\Vert _{X_m}.\) 
For the second part, note that \(\partial _{y_j}U_m^{-1}f= U_m^{-1}\partial _{y_j}f+\frac{1}{2}\partial _j\Psi U_m^{-1}f\)  and that
\(\begin{aligned}\Vert  U_m^{-1}\partial _{y_j}f\Vert _{X_m}^2 &= (\partial _{y_j}f,\partial _{y_j}f)_{L^2}^2\le c_\Sigma \int _{E} \langle \nabla _y f,\Sigma _n\nabla _y f\rangle + \frac{1}{4} \langle y,\Sigma _n y\rangle f^2\,\mathrm {d}(x,y)\\&\le \frac{1}{4}\left( a(-L_{n,m}^{\prime }f,f)_{L^2}+b_n\Vert f\Vert _{L^2}^2 \right) \le \frac{1}{4} D_1^2n^{2\beta }\Vert (I-L_{n,m})U_m^{-1}f\Vert ^2_{X_m}.\end{aligned}\) 


The remainder of the proof of thm:ess-m-diss regarding essential m-dissipativity now follows analogously (with \(\beta <\alpha <\frac{1}{\gamma }\) ) to the original proof of [1]}. By (REF ), \((Lf,1)_X=0\)  for all \(f\in \mathcal {C}\) , so \(\mu \)  is invariant for \((L,\mathcal {C})\) , which is an abstract diffusion operator. Then by [12]}, this implies that the generated semigroup is sub-Markovian. Invariance of \(\mu \)  for \((L,\mathcal {C})\)  also immediately implies invariance of \(\mu \)  for \((T_t)_{t\ge 0}\) .
By considering \(\widetilde{Q}\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =-Q\( instead of \) Q\( we obtain the same result for \) L=S+A\(,which coincides with the adjoint \) (L*,D(L*))\( of \) L\( on \) C\(.Hence \) (L*,D(L*))\( is the closure of \) (S+A,C)\( and generates the adjoint semigroup \) (Tt*)t0\(,for which \)\( is also invariant. Now conservativity of \) (Tt)t0\( is immediate, which concludes the proof of {thm:ess-m-diss}.\)

The weak hypocoercivity framework
We briefly summarize the weak hypocoercivity framework as well as the main result from [2]}, which along with [14]} is recommended for more details.
Let \((H,(\cdot ,\cdot ),\Vert \cdot \Vert )\)  be a separable Hilbert space with orthogonal decomposition \(H=H_1\oplus H_2\)  given by the orthogonal projection \(P:H\rightarrow H_1\) . Let \((L,D(L))\)  be a densely defined linear operator generating a \(C_0\) -semigroup \((T_t)_{t\ge 0}\)  on \(H\)  and let \(D\)  be a core of \((L,D(L))\) . Further assume that \(L\)  decomposes into \(L=S-A\)  on \(D\) , where \(S\)  is symmetric and \(A\)  is antisymmetric. This implies that \((S,D)\)  and \((A,D)\)  are closable; we denote their closures by \((S,D(S))\)  and \((A,D(A))\) , respectively. We assume the following:
Assumption (WH1)  
\(H_1\subset \mathcal {N}(S)\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) ={fD(S) : Sf=0}\(, that is \) H1D(S)\( and \) SP=0\(, which implies \) (I-P)DD(S)\(.\)
Assumption (WH2)  
\(PD\subset D(A)\)  and \(PAP|_{D}=0\) .

Under these assumptions, \((PA,D(A))\)  is closable with closure \((PA,D(PA))\)  and \(AP\)  with domain \(D(AP)\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) ={fH:PfD(A)}\( is closed and densely defined with \) (AP)*=-PA\( on \) D(A)\(. By von Neumann^{\prime }s theorem, the operators \) G=-(AP)*AP\( and \) I-G\( with domain \) D(G)=D((AP)*AP)={fD(AP):APfD((AP)*)}\( are self-adjoint and the latter admits a bounded linear inverse. We define the operator \) B\( with domain \) D(B)=D((AP)*)\( via\begin{equation}B\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\end{equation}=(I+(AP)^*AP)^{-1}(AP)^*.\) 
Then \(B\)  is bounded and extends to a bounded operator \((B,H)\)  with \(\Vert B\Vert \le 1\)  and \(PB=B\) . This allows us to formulate:
Assumption (WH3)  
We assume \(D\subset D(G)\)  and that there is a constant \(N<\infty \)  such that
\(\begin{aligned}(BS(I-P)f,Pf)_H&\le \frac{N}{2}\Vert (I-P)f\Vert _H\Vert Pf\Vert _h,\\-(BA(I-P)f,Pf)_H&\le \frac{N}{2}\Vert (I-P)f\Vert _H\Vert Pf\Vert _h, &&f\in D.\end{aligned}\) 


Note that if \((AP)D\subset D(A)\) , then \(D\subset D(G)\)  with \(G=PAAP\)  on \(D\) .
In order to ascertain the inequalities in ass:hypoc-3 later for our application,
we use the following Lemma, see [1]}:
Lemma 3.1 
Assume that \(D\)  is a core of \((G,D(G))\) .
Let \((T,D(T))\)  be a linear operator with \(D\subset D(T)\)  and assume \(AP(D)\subset D(T^*)\) .
Then
\((I-G)(D)\subset D((BT)^*) \quad \text{ with }\quad (BT)^*(I-G)f = T^*APf,\quad f\in D.\) 
If there exists some \(C<\infty \)  such that
\(\Vert (BT)^* g\Vert  \le C\Vert g\Vert \qquad \text{ for all }g\in (I-G)D\) 
then \((BT,D(T))\)  is bounded and its closure \((\overline{BT})\)  is a bounded operator on \(H\)  with
\(\Vert  \overline{BT}\Vert =\Vert  (BT)^*\Vert \) .
In particular, if \((S,D(S))\)  and \((A,D(A))\)  satisfy these assumptions,
the corresponding inequalities in ass:hypoc-3
are satisfied with \(N=2\max \lbrace \Vert  (BS)^*\Vert , \Vert  (BA)^*\Vert \rbrace \) .

Now let \(\Theta :H\rightarrow [0,\infty ]\)  be a functional such that \(\lbrace f\in H:\Theta f<\infty \rbrace \)  is dense in \(H\) .
Denote by \((\mathrm {e}^{tG})_{t\ge 0}\)  the \(C_0\) -semigroup generated by \(G\) . We assume the following:
Assumption (WH4)  
For each \(f\in H\)  and \(t\ge 0\) , the functional \(\Theta \)  satisfies
\(\Theta (T_tf)\le \Theta (f), \quad \Theta (\mathrm {e}^{tG}f)\le \Theta (f), \quad \Theta (Pf)\le \Theta (f).\) 


Assumption (WH5)  
For any \(f\in D(L)\)  there is some sequence \((f_n)_{n\in \mathbb {N}}\)  in \(D\)  such that \(f_n\rightarrow f\)  in \(H\)  and
\(\limsup _{n\rightarrow \infty }(-Lf_n,f_n)_H\le (-Lf,f)_H,\quad \limsup _{n\rightarrow \infty }(\Theta f_n)\le \Theta (f).\) 


Finally, we assume the following weak Poincaré inequalities to hold for \(S\)  and \(A\) :
Assumption (WH6)  
There exist decreasing functions \(\alpha _i:(0,\infty )\rightarrow [1,\infty )\) , \(i=1,2\) , such that
\(\Vert Pf\Vert ^2\le \alpha _1(r)\Vert APf\Vert ^2+r\Theta (Pf),\quad r>0,\ f\in D(AP)\) 
and
\(\Vert (I-P)f\Vert ^2\le \alpha _2(r)(-Sf,f)_H+r\Theta (f),\quad r>0,\ f\in D.\) 


This allows us to state the main weak hypocoercivity result:
Theorem 3.2 
Let ass:hypoc-1-ass:weak-poincare be satisfied. Then there exist constants \(c_1,c_2>0\)  such that
\(\Vert T_tf\Vert ^2\le \xi (t)(\Vert f\Vert ^2+\Theta (f)),\quad t\ge 0,\ f\in D(L)\) 
holds for
\(\xi (t)\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) 
=c1{ r>0:c2 t1(r)22( r1(r)2(1r) ) },

which goes to 0 as \(t\rightarrow \infty \) .

Weak hypocoercivity for generalized stochastic Hamiltonian systems with multiplicative noise
We now embed the differential operator \(L\)  as defined in def:operators and its associated semigroup into the weak hypocoercivity framework.
We define
\(H\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) 
={ fX: (f)=0}X,   H1={ fH: f(x,y) does not depend on y},
\(as well as\) 
P:HH1, Pf=Rd2f(x,y) 2(dy)    for fH
\(and\) 
D={ fC(Rd1+d2):f has compact support, (f)=0 }H.
\(Note that each \) 
D\( admits the representation \) f=g-(g)\( for some \) gCc(Rd1+d2)\(.Simple approximation shows that the closures of \) (S,C)\(, \) (A,C)\( and \) (L,C)\( in \) X\( act trivially on constants,which suggests the natural definition \) Sf=Sg\( for \) fD\(, similarly for the other operators.\) Let \((L,D(L))\) , \((S,D(S))\)  and \((A,D(A))\)  be the closures of the dissipative operators \((L,D)\) , \((S,D)\)  and \((A,D)\)  in \(H\)  respectively.
Under the assumptions of thm:ess-m-diss, \((L,C_c^\infty (\mathbb {R}^{d_1+d_2}))\)  is essentially m-dissipative in \(X\) , which implies essential m-dissipativity of \((L,D)\)  in \(H\) .
Indeed, let \(f\in H\) . Then there exists a sequence \((g_n)_{n\in \mathbb {N}}\)  in \(\mathcal {C}\)  such that \((I-L)g_n\rightarrow f\)  in \(X\)  as \(n\rightarrow \infty \) .
Define \(f_n\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =gn-(gn)\( for \) nN\(, then \) fnD\( for all \) nN\( and \) Lfn=Lgn\(.Hence \) (I-L)fn=(I-L)gn-(gn)f\( in \) H\( as \) n\(, since \) (f)=0\(.\) We can therefore start verifying the conditions given in sec:weak-hypoc.
Note that Condition ass:hypoc-1 is fulfilled by definition, see [2]}.
Since the operator \((A,D)\)  coincides with the corresponding operator in [2]},
the proofs which only concern \(A\)  can be repeated verbatim.
We summarize those results after assuming the following regularity for the potentials:
Assumption (\(\Phi \) 3)  
We assume that \(\Phi \in C^2(\mathbb {R}^{d_1})\)  and that there is a constant \(C<\infty \)  such that
\(|\nabla ^2\Phi (x)|\le C(1+|\nabla \Phi (x)|) \quad \text{ for all }x\in \mathbb {R}^{d_1}.\) 


Assumption (\(\Psi \) 5)  
There is some \(\psi \in C^2(\mathbb {R})\)  such that \(\Psi (y)=\psi (|y|^2)\) .

Note that due to ass:x-potential-3, we can apply thm:arbitrary-integrability
to \(V=\Phi \)  with \(g\equiv 1\)  to obtain an analogue to rem:psi-lp.
We collate this in the following:
Corollary 4.1 
Let \(\Phi \)  satisfy ass:x-potential-3.
Then \(|\nabla \Phi |\in L^p(\mu _1)\)  and \(|\nabla \Psi |\in L^p(\mu _2)\)  for any \(1\le p<\infty \) .
The same integrability also extends to the Hessian matrices \(|\nabla ^2\Phi |\)  and \(|\nabla ^2\Psi |\)  as well.

We can now summarize the results for the antisymmetric part \(A\)  proved in [2]}:
Lemma 4.2 If ass:x-potential-3 and ass:y-potential-5 are satisfied, then
Condition ass:hypoc-2 is fulfilled and \((AP)D\subset D(A)\) .
Furthermore, the operator \((G,D)\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =(PAAP,D)\(, which is given for all \) fD\( by\)\(Gf= \frac{\mu _2(|\nabla \Psi |^2)}{d_2}\sum _{i,j=1}^{d_1} (QQ^*)_{ij} (\partial _{x_j}\partial _{x_i}-\partial _{x_j}\Phi \,\partial _{x_i})Pf,\)\(is essentially self-adjoint and its closure \) (G,D(G))\( generates a sub-Markovian strongly continuous semigroup on \) H\(.Moreover, there is a constant \) cA\( only depending on the choice of \)\( and \)\( such that\)\(\Vert (BA)^*g\Vert _H\le c_A\Vert g\Vert _H\qquad \text{ for all }g\in (I-G)D.\)\(\) 
This allows us to use lem:auxiliary-bound to prove ass:hypoc-3.
Indeed, the inequality for the antisymmetric part follows immediately from the previous Lemma, so it remains to show the first inequality. For this, we introduce the final assumption on \(\Psi \) :
Assumption (\(\Psi \) 6)  
\(\Psi \)  is three times weakly differentiable and for any \(1\le i,j,k\le d_2\) , it holds that
\(\partial _i\partial _j\partial _k\Psi \in L^2(\mu _2)\) .

Note that in light of cor:arbitrary-integrability, this holds in particular if the third order partial derivatives are dominated by a multiple of \((1+|\nabla ^2\Psi |^\beta )\)  for any \(0<\beta <\infty \) .
We also need to assume some additional integrability on \(\nabla \Sigma \) :
Assumption (\(\Sigma \) 4)  
There is some \(1<p_\Sigma \le \infty \)  such that \(|\nabla \Sigma |\in L^{2p_\Sigma }(\mu _2)\) .

Lemma 4.3 
Let ass:y-potential-5, ass:y-potential-6 and ass:mat-cond-4 be satisfied.
Then the operator \(T:D\rightarrow L^2(\mu )\)  with

Tf=i=1d1 wi(xi Pf)
       for wi:Rd2R    defined by 
wi=j,k,=1d2qij (
k ak(j)
+ ak(kj)
- ak(j)(k)
)

is well-defined and there is some \(R<\infty \)  such that \(\Vert Tf\Vert _{L^2(\mu )}^2\le R\Vert  \nabla f_P\Vert _{L^2(\mu _1)}^2\) 
for all \(f\in D\) .

Proof 7 It holds that \(\Vert Tf\Vert _{L^2(\mu )}^2\le \sum _{i=1}^{d_1}\Vert w_i\Vert _{L^2(\mu _2)}^2 \Vert \nabla f_P\Vert _{L^2(\mu _1)}^2\) 
and for each \(1\le i\le d_1\) :
\(\Vert w_i\Vert _{L^2(\mu _2)}\le |Q| \sum _{j,k,\ell =1}^{d_2}\Vert \partial _k a_{k\ell }(\partial _{\ell }\partial _j\Psi )\Vert _{L^2}+ \Vert a_{k\ell }(\partial _k\partial _{\ell }\partial _j\Psi )\Vert _{L^2}+ \Vert a_{k\ell }(\partial _{\ell }\partial _j\Psi )(\partial _k\Psi )\Vert _{L^2}\) 
Let \(1\le q_\Sigma <\infty \)  be such that \(\frac{1}{p_\Sigma }+\frac{1}{q_\Sigma }=1\) .
Then by ass:mat-cond-4 and cor:arbitrary-integrability, it holds that
\(\Vert \partial _k a_{k\ell }(\partial _{\ell }\partial _j\Psi )\Vert _{L^2}\le \Vert (\partial _k a_{k\ell })^2\Vert _{L^{p_\Sigma }} \Vert (\partial _\ell \partial _j\Psi )^2\Vert _{L^{q_\Sigma }}< \infty .\) 
Boundedness of \(\Sigma \)  together with ass:y-potential-6 yields
\(\Vert a_{k\ell }(\partial _k\partial _{\ell }\partial _j\Psi )\Vert _{L^2}\le M_\Sigma \Vert \partial _k\partial _{\ell }\partial _j\Psi \Vert _{L^2}<\infty ,\) 
and by ass:y-potential-3 and again cor:arbitrary-integrability, we also get
\(\Vert a_{k\ell }(\partial _{\ell }\partial _j\Psi )(\partial _k\Psi )\Vert _{L^2}\le KM_\Sigma \left( \Vert \partial _k\Psi \Vert _{L^2} + \Vert |\nabla \Psi |^{1+\alpha }\Vert _{L^2} \right)<\infty .\) 
This shows that each \(w_i\)  is in \(L^2(\mu _2)\)  and therefore \(R\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =i=1d1wiL2(2)2<\(.\) 
Now we show that \((AP)D\subset D(S^*)\) .
To that end, let \(f\in D\) , \(h\in D(S)\)  and \((h_n)_{n\in \mathbb {N}}\)  be a sequence in \(D\)  such that \(h_n\rightarrow h\)  and \(Sh_n\rightarrow Sh\)  in \(H\)  as \(n\rightarrow \infty \) .
Then integration by parts applied twice yields
\(\begin{aligned}(Sh,APf)_H&=\lim _{n\rightarrow \infty }(Sh_n,-Q\nabla \Psi \cdot \nabla (Pf))_H=\lim _{n\rightarrow \infty }\mu (\langle \nabla _y h_n,\Sigma \nabla _y(Q\nabla \Psi \cdot \nabla _x(Pf))\rangle )\\&= \lim _{n\rightarrow \infty } (h_n, Tf)_{L^2(\mu )}= (h,Tf)_{L^2(\mu )}.\end{aligned}\) 
Setting \(h\equiv 1\)  shows that \(Tf\in H\)  since \(Sh=0\) , which implies \(APD\subseteq D(S^*)\) .
Now by lem:auxiliary-bound, it follows that \((BS)^*(I-G)f=S^*APf=Tf\)  and
\(\Vert Tf\Vert _H^2\le R \Vert \nabla f_P\Vert _{L^2(\mu _1)}^2\le \frac{d_2R|Q^{-1}|^2}{\mu _2(|\nabla \Psi |^2)} \frac{\mu _2(|\nabla \Psi |^2)}{d_2} \Vert Q^*\nabla f_P\Vert _{L^2(\mu _1)}^2.\) 
Further, since via integration by parts
\(\begin{aligned}\Vert Q^*\nabla f_P\Vert _{L^2(\mu _1)}^2&= -\int _{\mathbb {R}^{d_1}} \sum _{i,j=1}^{d_1} (QQ^*)_{ij}\, f_P\,(\partial _j\partial _if_P-\partial _j\Phi \partial _i f_P)\ \mathrm {d}\mu _1\\&= -\int _{\mathbb {R}^{d_1+d_2}} \sum _{i,j=1}^{d_1} (QQ^*)_{ij}\, Pf\,(\partial _{x_j}\partial _{x_i}Pf-\partial _j\Phi \partial _{x_i}Pf)\ \mathrm {d}\mu \\&= -\frac{d_2}{\mu _2(|\nabla \Psi |^2)}\int _{\mathbb {R}^{d_1+d_2}} Pf Gf\,\mathrm {d}\mu \end{aligned}\) 
we obtain for \(g\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =(I-G)f\( due to dissipativity of \) (G,D)\( that\)\(\begin{aligned}\Vert (BS)^*g\Vert _H^2&=\Vert Tf\Vert _H^2\le \frac{d_2R|Q^{-1}|^2}{\mu _2(|\nabla \Psi |^2)} \int _E Pf\, (-G)f\ \mathrm {d}\mu \\&\le \frac{d_2R|Q^{-1}|^2}{\mu _2(|\nabla \Psi |^2)} \Vert Pf\Vert _H (\Vert (I-G)f\Vert _H + \Vert f\Vert _H) \\&\le \frac{2d_2R|Q^{-1}|^2}{\mu _2(|\nabla \Psi |^2)} \Vert (I-G)f\Vert _H^2= \frac{2d_2R|Q^{-1}|^2}{\mu _2(|\nabla \Psi |^2)} \Vert g\Vert _H^2.\end{aligned}\)\(Using the second part of {lem:auxiliary-bound}, this shows that \) (BS,D(S))\( is bounded and hence {ass:hypoc-3} is fulfilled.\)
Definition 4.4 Let the functional \(\Theta :H\rightarrow [0,\infty ]\)  be defined by
\(\Theta f\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) 
=fosc2=(ess supf-ess inff)2
\(for all \) 
H\(.\) 
Then the set \(\lbrace f\in H:\Theta f<\infty \rbrace \supset D\)  is dense in \(H\)  and ass:hypoc-4 is satisfied due to \(L^\infty \) -contractivity of the semigroups.
We show ass:hypoc-5 using the same construction as in [2]}: Fix some \(f\in D(L)\)  and set \(\gamma _1\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =ess inff\(, \) 2=ess supf\(.Let \) (gn)nN\( be a sequence in \) D\( such that \) gnf\( and \) LgnLf\( in \) H\( as \) n\(. Then set \) fn=hn(gn)\(, where \) hnC(R)\( satisfies\) 0hn'1\( and\)\(h_n(r)={\left\lbrace \begin{array}{ll}r & \text{ for }r\in [\gamma _1,\gamma _2]\\\gamma _1-\frac{1}{2n} & \text{ for }r\le \gamma _1-\frac{1}{n}\\\gamma _2+\frac{1}{2n} & \text{ for }r\ge \gamma _2+\frac{1}{n}\end{array}\right.}.\)\(Then \) fnf\( in \) H\( as \) n\(, \) nfnoscfosc\(, and\)\(\begin{aligned}\limsup _{n\rightarrow \infty } (-Lf_n,f_n)&=\limsup _{n\rightarrow \infty } \mu (\langle \nabla _y f_n,\Sigma \nabla _y f_n\rangle )=\limsup _{n\rightarrow \infty } \mu ((h_n^{\prime }(g_n))^2 \langle \nabla _y g_n,\Sigma \nabla _y g_n\rangle )\\&\le \limsup _{n\rightarrow \infty } \mu (\langle \nabla _y g_n,\Sigma \nabla _y g_n\rangle )=\limsup _{n\rightarrow \infty } (-Lg_n,g_n)= (-Lf,f).\end{aligned}\)\(Finally, it remains to show the weak Poincaré inequalities are satisfied. For this, we use \cite [Theorem 3.1]{RW01}, which for our purposes states that, givena probability measure \) V=eVdx\( on \) Rd\( with locally bounded \) V\(, there exists a decreasing function \) :(0,)(0,)\( such that\begin{equation}\mu _V(f^2)\le \alpha (r)\mu _V(|\nabla f|^2)+r\Vert f\Vert _\mathrm {osc}^2,\quad r>0,f\in C_b^1(\mathbb {R}^d).\end{equation}The first inequality in {ass:weak-poincare} is already proved in \cite {GW19}, which also gives a procedure to show the second one:Let \) fD\( and \) xRd1\(, set \) fx=f(x,)-Pf(x)C(Rd2)\(.Then \) fx\( has compact support, so that \) fx\( is bounded; \) 2(fx)=0\( and \) fxoscfosc\(.Therefore, (\ref {eq:wp-base}) is applicable and yields the existence of a decreasing function \)\( such that\)\(\begin{aligned}\mu _2(f_x^2)&\le \alpha _\Psi (r)\mu _2(|\nabla _y f(x,\cdot )|^2)+r\Vert f_x\Vert _\mathrm {osc}^2\\&\le c_\Sigma \alpha _\Psi (r)\mu _2(\langle \nabla _y f(x,\cdot ),\Sigma \nabla _y f(x,\cdot )\rangle )+r\Vert f\Vert _\mathrm {osc}^2\end{aligned}\)\(for all \) r>0\(, \) fD\( and \) xRd1\(.Integrating that expression wrt.~\) 1\( gives\)\(\begin{aligned}\Vert (I-P)f\Vert _H^2&=\int _{\mathbb {R}^{d_1}}\mu _2(f_x^2)\,\mu _1(\mathrm {d}x)\le c_\Sigma \alpha _\Psi (r) \int _E \langle \nabla _y f,\Sigma \nabla _y f\rangle \,\mu (\mathrm {d}(x,y))+r\Vert f\Vert _\mathrm {osc}^2\\&= c_\Sigma \alpha _\Psi (r) (-Sf,f)_H +r\Theta f\end{aligned}\)\(for all \) fD\(, \) r>0\(, so the second inequality in {ass:weak-poincare} is satisfied with \) 2=c\(.\begin{proof}[of {thm:main-result}]Let the assumptions of {thm:main-result} hold.Then clearly all assumptions of {thm:ess-m-diss} are satisfied (see {cor:arbitrary-integrability}), which yields the existence of the C_0-semigroup (T_t)_{t\ge 0} on L^2(\mu ) with the stated properties. The existence of the decreasing functions \alpha _\Phi  and \alpha _\Psi  is again provided by \cite [Theorem 3.1]{RW01}.\end{proof}In order to satisfy {ass:y-potential-5}, we first shift the \) y\(-variable by \) a\( and then consider \)(y)=(-1y)=(|y|2)\(, with \)2\( and \)\( defined accordingly. Let \) J:L2()L2()\( be given by \) Jf(x,y)=|()|-1f(x,-1y)\(.Then \) J\( is a unitary transformation which leaves \) D\( invariant, and we define the operator \) (L,D)\( by \)L=JLJ-1\(.This operator has the same structure as \) L\(, except with \)Q=Q*\( and \)(y)=(-1y)*\(.So wlog we can assume that already \) (y)=(|y|2)\(. We then restrict the setting to \) HX\( and consider the closure \) (LH,D(LH))\( of \) (L,D)\( in \) H\(with the corresponding semigroup \) (TtH)t0\( on \) H\(.\) It can then easily be checked with the previous considerations in this section that all conditions for thm:abstract-hypoc are met, which gives the existence of constants \(0<\tilde{c}_1,c_2<\infty \)  such that
\(\mu ((T_t^H f)^2)\le \tilde{c}_1\xi (t)(\mu (f^2)+\Vert f\Vert _{\mathrm {osc}}^2),\qquad t\ge 0,\ f\in D(L^H)\) 
with \(\xi \)  given as in the statement of thm:main-result.
Since \(L\)  and \(L^H\)  coincide on \(D\) , the semigroup \((T_t)_{t\ge 0}\)  coincides with \((T_t^H)_{t\ge 0}\)  on \(H\) .
Invariance and conservativity imply that
\(T_t(\mu (f))=\mu (f)T_t(1)=\mu (f)=\mu (T_tf)\quad \text{ for all }f\in L^2(\mu ).\) 
Since \(f-\mu (f)\in D(L^H)\)  for all \(f\in D(L)\) , this yields
\(\mu ((T_tf)^2)-\mu (T_tf)^2=\mu ((T_t^H(f-\mu (f)))^2)\le \tilde{c}_1\xi (t)(\mu ((f-\mu (f))^2)+\Vert f\Vert _{\mathrm {osc}}^2)\) 
for all \(f\in D(L)\)  and \(t\ge 0\) .
Since \(|f-\mu (f)|\le \Vert f\Vert _\mathrm {osc}\) , this shows the wanted estimate with \(c_1=2\tilde{c}_1\)  for all \(f\in D(L)\) .
This extends to \(f\in L^\infty (\mu )\)  by approximation as in the proof of ass:hypoc-5.

Applications
We briefly present direct applications of the hypocoercivity result to PDEs and SDEs.
Since the principles used are the same, we refer to [1]} for details.
We assume the conditions stated in thm:main-result.
The abstract Cauchy problem
We consider the abstract Cauchy problem associated with the operator \(L\)  on \(X=L^2(\mu )\) :
\(u(0)=u_0, \qquad \partial _t u(t) = Lu(t), \quad \text{ for all }t\ge 0,\) 
where \((L,D(L))\)  is the closure of \((L,\mathcal {C})\)  on \(X\) , \(u:[0,\infty )\rightarrow X\)  and \(u_0\in X\) .
Let \((T_t)_{t\ge 0}\)  denote the semigroup on \(X\)  generated by \((L,D(L))\) 
and set \(u(t)\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =Ttu0\(. Then \) u\( is the unique mild solution to (\ref {eq:cauchy-kol}) for all \) u0X\(and the unique classical solution for all \) u0D(L)\(, so in particular for all \) u0{ fC2(Rd1+ d2):f has compact support}\(.\) Furthermore, thm:main-result shows that for all bounded measurable \(u_0\) ,
we obtain convergence of the unique solution \(u(t)\)  to a constant as \(t\rightarrow \infty \) . More precisely, there are explicit constants \(c_1,c_2>0\)  depending on the choice of \(\Sigma \) , \(\Phi \)  and \(\Psi \)  such that for all \(t\ge 0\) ,
\(\left\Vert  u(t)-\int _E u_0\,\mathrm {d}\mu \right\Vert _X^2 \le c_1\xi (t) \left\Vert u_0 \right\Vert _\mathrm {osc}^2,\) 
with \(\xi \)  given by (REF ).

The associated Fokker-Planck equation in divergence form
Consider the Hilbert space \(\widetilde{X}\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =L2(E;)\(, where\)\(\widetilde{\mu }\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =Z()-12Z()-12
e(x)+(y) dx dy,
\(and define the differential operator \) 
LFP,C)\( on \) X\( for all \) fC=Cc(Rd1)Cc(Rd2)\( as in (\ref {eq:proto-fokker-planck}).The abstract Cauchy problem associated with the closure \) (LFP,D(LFP))\( is given by(\ref {eq:cauchy-fp}).Using the unitary transformation\)\(T:X\rightarrow \widetilde{X},\quad Tg=\rho g \quad \text{ with }\quad \rho (x,v)\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =e-(x)-(y),
\(as well as the result on the adjoint semigroup \) 
Tt*)t0\( stated at the end of the proof of {thm:ess-m-diss},we obtain essential m-dissipativity of \) (LFP, D)\(,where \) D=Cc2(E)\( is a \) T\(-invariant core of \) (L*,D(L*))\(.A simple approximation argument gives essential m-dissipativity of \) (LFP,C)\( on \) X\(, and the semigroup \) (Tt)t0=(T(Tt*)T-1)t0\( generated by the closure \) (LFP,D(LFP))\( induces solutions of the Cauchy problem as seen in {sec:cauchy}.\) Let \(u_0\in \widetilde{X}\)  such that \(\frac{u_0}{\rho }\)  is bounded and set \(u(t)\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =Tt u0\(. Then {thm:main-result} yields the existence of constants \) c1\(, \) c2\( depending on \)\(, \)\( and \)\( such that for all \) t0\(,\)\(\begin{aligned}\left\Vert  u(t)-\rho \int _E u_0\rho \,\mathrm {d}\widetilde{\mu } \right\Vert _{\widetilde{X}}^2&= \left\Vert  T(T_t^*)T^{-1}u_0-T(1)\int _E \frac{u_0}{\rho }\,\mathrm {d}\mu \right\Vert _{\widetilde{X}}^2\\&= \left\Vert  (T_t^*)T^{-1}u_0-\int _E T^{-1}u_0\,\mathrm {d}\mu \right\Vert _X^2\le c_1 \xi (t) \left\Vert \frac{u_0}{\rho } \right\Vert _\mathrm {osc}^2,\end{aligned}\)\(with \)\( given by (\ref {eq:conv-rate}). This shows convergence of the solution \) u\( to the stationary solution \) E u0 d(x,y)\( in \) X\( as \) t\( with the rate \)\(.\)

Degenerate Diffusion with multiplicative noise
Finally, we connect the convergence result to the initially stated Itô SDE ().
Under the assumptions of thm:ess-m-diss, the theory of Generalized Dirichlet Forms developed by Stannat (see [21]}) yields the existence of a Hunt process
\(\mathbf {M} = (\Omega ,\mathcal {F},(\mathcal {F}_t)_{t\ge 0},(X_t,Y_t), (P_{(x,y)})_{(x,y)\in \mathbb {R}^{d_1}\times \mathbb {R}^{d_2}})\) 
with state space \(E=\mathbb {R}^{d_1}\times \mathbb {R}^{d_2}\)  and \(P_{(x,y)}\) -a.s. continuous sample paths for all \((x,y)\in E\) ,
which is properly associated in the resolvent sense with the semigroup \((T_t)_{t\ge 0}\)  generated by \((L,D(L))\) .
In particular, we obtain that for each \(f\in \widetilde{X}\)  and all \(t>0\) , \(T_tf\)  is a \(\mu \) -version of \(p_tf\) , where \((p_t)_{t\ge 0}\)  is the transition semigroup of \(\mathbf {M}\)  given by
\(p_tf:\mathbb {R}^{d_1}\times \mathbb {R}^{d_2}\rightarrow \mathbb {R}, \qquad (x,y)\mapsto \mathbb {E}_{(x,y)}[f(X_t,Y_t)].\) 
Due to [22]}, for any \(\mu \) -probability density \(h\in L^1(\mu )\)  (including \(h\equiv 1\) ),
the probability law \(P_h\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =E P(x,y)h(x,y) d\( on the space of continuous paths solves the martingale problem for \) (L,D(L))\( on \) Cc2(E)\(, i.e. for each \) fCc2(E)\(, the stochastic process \) (Mt[f])t0\( given by\)\(M_t^{[f]}\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =f(Xt,Yt)-f(X0,Y0)-0t Lf(Xs,Ys) ds
\(is a martingale with respect to \) 
h\(. Additionally, by \cite [Lemma~2.1.8]{Conrad2011}, for \) fD(L)\( with \) f2D(L)\( and \) LfL4()\(, the process \) (Nt[f])t0\( defined by\)\(N_t^{[f]}\mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) =(Mt[f])2-0tL(f2)(Xs,Ys)-2(fLf)(Xs,Ys) ds,
   t0,
\(is also a martingale wrt.~\) 
h\(. Analysis of the coordinate processes together with Lévy^{\prime }s characterizationof Brownian motion shows that \) (Xt,Yt)t0\( is a weak solution to the SDE (\ref {eq:sde}), since \) =\( is invertible with locally bounded inverse.\) Under the assumptions of thm:main-result, we therefore obtain the existence of constants \(c_1,c_2\)  depending on \(\Sigma \) , \(\Phi \)  and \(\Psi \)  such that
\(\left\Vert  p_tf - \mu (f) \right\Vert _{L^2(E;\mu )}^2 \le c_1 \xi (t) \Vert f\Vert _{\mathrm {osc}}^2\) 
for all bounded measurable \(f:E\rightarrow \mathbb {R}\) . This implies that the law \(P_1\)  with initial distribution \(\mu \)  is strongly mixing, i.e. for any two Borel sets \(A_1,A_2\)  on the space of continuous paths, it holds
\(P_1(\varphi _t A_1\cap A_2) \rightarrow P_1(A_1) P_1(A_2)\quad \text{ as }t\rightarrow \infty ,\) 
where \(\varphi _tA_1\)  denotes the set of shifted paths
\(\varphi _t A_1 \mathrel {\hbox{\scriptsize .}\hbox{\scriptsize .}}\) 
={ (Zs)s0C([0,),E) (Zs+t)s0A1 }.
\(\) 
